Power at Resonance
Energy peak — wine glass physics. Builds on Driven Resonance.
Two identical tuning forks are placed near each other. You strike one and observe the other. One fork has a tiny piece of clay stuck to one prong, making it very slightly heavier (and therefore very slightly lower in frequency).
Compared to a pair of perfectly identical forks, the second fork (with clay) will:
Much smaller amplitude! The tuning fork has a very high Q factor (∼1000), which means its resonance peak is extremely narrow. Even a tiny frequency mismatch from the clay causes the driving frequency to fall off the peak, dramatically reducing the energy absorbed. This is why identical tuning forks demonstrate sympathetic resonance beautifully, but even slightly mismatched ones barely respond.
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Watch the power absorption curve as you adjust Q. At high Q, the peak is tall and narrow — only a tiny range of frequencies gets significant power. At low Q, the peak is broad and squat — a wider range of frequencies can drive the system.
Q = ω₀ / Δω = ω₀ / (2γ)
Q also equals 2π × (energy stored / energy lost per cycle). A high-Q system rings for many cycles; a low-Q system dies quickly. A tuning fork has Q ∼ 1000; a wet sponge has Q ∼ 1.
Quartz crystal oscillators in watches have Q ∼ 10,000–100,000. This extreme selectivity keeps time accurate to seconds per month. The crystal only vibrates at its exact resonant frequency.
Optical cavities in lasers have Q factors of 10⁸ or higher. Only photons at exactly the cavity’s resonant frequency survive, producing the laser’s incredibly pure single-frequency light.
A fine crystal wine glass has Q ∼ 1000. Flick it and it rings for several seconds. That long ring-down means a narrow resonance peak — only a very precise frequency can shatter it.
Buildings have natural frequencies. Engineers design structures so their Q factor at earthquake frequencies is low (high damping), spreading the resonance peak to avoid catastrophic amplification at any single frequency.
“The Q factor is nature’s dial between selectivity and robustness. High Q: exquisitely selective but fragile. Low Q: broadly responsive but dilute. Every oscillator in the universe sits somewhere on this dial.”
Power Resonance Lab
Explore how Q factor shapes the resonance curve. Adjust damping and watch the power absorption peak morph from a razor-sharp spike to a broad hill. Measure the FWHM, peak power, and verify Q = ω₀/Δω.
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Set Q to about 5 and sweep the driving frequency — you’ll see a gentle hill of power absorption. Now increase Q to 50 — the peak shoots up and narrows. At Q = 500, only a tiny frequency range absorbs significant power. Watch the phase plot too: at low Q the phase changes gradually; at high Q it flips almost instantaneously from 0° to 180° right at resonance. Measure the peak width at half-maximum and verify it equals ω₀/Q.
Q factor connects the time domain (how long does the oscillator ring?) to the frequency domain (how narrow is the resonance peak?). A system that rings for N cycles has Q ≈ πN, and its resonance peak width is ω₀/Q. This is the essence of the Fourier uncertainty principle: you cannot have a short ring AND a narrow peak.